If we restrict the consumption at $\mathbb R_+^n$, then it seems like we are implicitly assuming that the Engel curve pass through the origin, so the iff condition would be homothetic preference.

However, if we let $x\in\mathbb R^n$, other classes of preference would also give linear Engel curves. For example, those represented by a quasi linear utility in two dimensions: $u(x)=x_1+f(x_2)$; in this case, the Engel curves are parallel lines.

It could be assumed that, as in the standard consumer theory, the preference is non-satiated and continuous, so the indifference sets are curves.


Are negative incomes (perhaps even negative prices) also allowed? Unless they are, non-horizontal or non-vertical lines are impossible as Engel curves because they have to intersect the negative quadrant of the coordinate system. I will come back to this later.

Without additional restrictions (perhaps a lot of additional restrictions) on the preference relations, for any non-negative sloped line I can fabricate a utility function such that the line will be its Engel curve.

Let the line $L$ consist of the points $((x,y)\in\mathbb{R}^2|c = a \cdot x + b \cdot y)$ where $a,b,c \in \mathbb{R}$. I will denote this set of points by $L$, the line. It is assumed that we are talking about an actual line, so $a,b$ are not simultaneously equal to $0$.

Let $$ U(x,y) = \left\{\begin{array}{lc} \arctan (b \cdot x + a \cdot y) & \text{ if } (x,y) \in L \\ -\pi & \text{ if } (x,y) \notin L. \end{array} \right. $$

Claim The utility function will always be maximized by a point on $L$.

This is easy, because $\arctan$ maps to $(-\frac{\pi}{2},\frac{\pi}{2})$, thus its points always yield greater utility than other points. We specified that the slope of $L$ is non-negative, so for any positive price pair the budget line will cross it, enabling us to chose a point on $L$.

We now come back to the question of whether income $I$ can be negative. If they can be, than every point $L$ will be a solution of the consumers utility maximization problem for a certain $I,p_x,p_y$. The reason for this is that $b \cdot x + a \cdot y$ creates an ordering of the points of $L$. (Lines perpendicular to $L$ have a slope of $-b/a$, these are the level curves of $b \cdot x + a \cdot y$.) Select a point $(x_0,y_0) \in L$. By setting $I$ to $$ I = p_x \cdot x_0 + p_y \cdot y_0, $$ the consumer has barely enough money to purchase this basket. If $L$ is non-negative sloped and prices are positive then no points with higher utility are attainable.

The construction I gave is not unique. One could also use the continuous utility function $$ \hat{U}(x,y) = \min\left(a \cdot \left(x - \frac{c}{a} \right); - b \cdot y\right) $$ to achieve the same result. (If $a = 0$ one may use $$ \hat{U}(x,y) = \min\left(a \cdot x; - b \cdot \left(y - \frac{c}{b} \right)\right) $$ instead.)

  • $\begingroup$ Thanks for your help answer! I might start another question with additional restrictions. What are some of the reasonable yet not-too-strong assumptions? $\endgroup$
    – High GPA
    Jun 9 '19 at 15:36
  • 2
    $\begingroup$ @HighGPA I have no clue. I don't think this is a much studied problem, the Engel-curve seems to be an intro to Micro tool, and in that context people usually study non-negative good baskets. $\endgroup$
    – Giskard
    Jun 9 '19 at 15:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.